Full text of "Analytical Mechanics"

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obtain the curve which represents the resultant motion. In Fig. 135 the curves (I) and (IT) represent the component motions and curve (III) represents the resultant motion.
FIG. 135.
(2) Draw two concentric auxiliary circles with radii equal to the amplitudes of the component motions; draw a radius in each circle making an angle with the Z-axis, equal to the phase angle of the corresponding motion; the vector sum of these radii gives the radius of the auxiliary circle for the resultant motion and the corresponding phase angle. By the help of this auxiliary circle the displacement-time curve of the resulting motion can be drawn without drawing those of the component motions.
PROBLEMS.
Find the resultant motion due to the superposition of two motions defined by the following pairs of equations:
(1)  Xi = ai sin coŁ   and   Ł2 = a2sin( u>t + -}• >(2)  xi = ai sin coŁ   and   x* — a<> cos f at — - j •
t(3)  xi = ai cos cot   and   z2 = a2 cos f ut +1V
'(4)  Xi = ai sin coŁ   and   x2 = a2 sin (coŁ +•§). (5)  Xi = di sin ait   and   x% — a* cos cci.```